To begin the discussion of the background of the invention, certain terms will be defined, and illustrated in connection with a concrete example of an industrial process. A more elaborate discussion of the terms defined here, and a detailed analysis of the field of design of experiments for industrial processes, may be found in the treatise "System of Experimental Design", by Genichi Taguchi, White Plains, N.Y. and Dearborn, Mich.: UNIPUB/Kraus International Publications (1987).
Industrial processes have a wide variety or objectives. A common objective is to mass-produce a product. For instance, an industrial process could be for the manufacture are magnetic heads.
In an industrial process, there will likely be various reactors which influence the outcome are the process. For instance, in the magnetic head manufacturing process, various factors could include the desired current density the head can carry, a temperature for a chemical bath in which the heads being manufactured are immersed, the chemical makeup of the bath, the composition of the head material, the magnetic field the head is to produce, the physical dimensions of the head, and the gap size.
For each factor, various levels can be set. For instance, the bath temperature factor of the head manufacturing process given above could have levels such as 27.degree., 33.degree., or 45.degree. Celsius.
The outcome of the industrial process, i.e., the quality of the final product, depends on the levels set for the various factors. In designing an industrial process, therefore, it is important to identify levels for the factors which lead to optimal or near-optimal results.
Depending on the nature of a given factor in an industrial process, variation in the level for the factor may or may not have a significant effect on the result of the process. If the level of a factor does have a significant effect, the effect is called a "main effect". Therefore, for those factors for which level variation produces a main effect, it is particularly important to select the level with the objective of optimal result in mind.
Moreover, there is a certain degree of variation in the levels for the factors in the actual process, which is impracticable to control, it is also desirable to identify levels for which such variation does not drastically reduce the quality of the output. Taguchi analyzed this variation, and used the expression "signal-to-noise ratio" (SN ratio) to refer to a measurement which is proportional to the reciprocal to the error variance. Since the error variance should be as small as possible, the SN ratio should be as large as possible.
It is also often the case that combinations of two or more factors provide a synergistic significant influence on the outcome of the process. The effect on the outcome produced by such a combination of levels for factors is called an interaction. In general, an interaction based on the levels for p+1 factors is called a p-th order interaction. Therefore, for those combinations of factors for which level variations produce an interaction, it is also important to select their levels with the objective of optimal result in mind.
When an industrial process is being developed, levels for the various factors are conventionally determined empirically. That is, experiments are run in which various levels are set for the factors, and the outcomes of the experiments are compared to determine which levels for the factors produce optimal results. Experiments are defined in terms of levels for each of the factors. The levels are selected based on criteria such as prior knowledge of a general range of levels which are likely to produce near optimal results. The selection of levels outside such a range will increase the number of experiments, without adding appreciably to the value of the experimental results. However, it is important to include levels within the desired range, so that results will give a good indication of where, within the desired range, the best level lies. After the experiments have been run, the results are compared to determine which levels produced the best results.
There may be a single combination of levels which produces results much better than any other combination of levels, or there may be a number of combinations which produce comparably good results. That is to say, the SN ratio varies from one industrial process to another. Where changing levels produces relatively little change in results, other criteria such as cost or convenience may be used to determine which combination of levels should be used. When levels which lead to optimal results are identified from the experimental results, the factors are set to those levels, and mass production is begun.
Taguchi provided a method for defining experiments using orthogonal arrays. The use of orthogonal arrays provides a method for representing a set of experiments which is convenient to create and manipulate. An orthogonal array is, essentially, a two-dimensional array of level symbols, in which one dimension (i.e., rows) represents a set of industrial process experiments to be performed, and the other dimension (i.e., columns) represents the factors of the process whose levels are to be manipulated in the experiments being defined. Each element of the array, therefore, is a level for a given one of the factors in a given one of the experiments. While levels can be expressed in terms of their own particular values and units of measure. Taguchi expressed in terms of level symbols. In particular, Taguchi used, as level symbols, integers starting with one and ascending.
An example of an orthogonal array is provided in FIG. 1, which is a reproduction of Table 6.1 on page 166 of Taguchi. FIG. 1 defines eight experiments, designated by the numbers 1 through 8 along the left column. The experiments are defined in terms of seven factors, designated 1 through 7 along the top row. For each position of the orthogonal array, a level for a given factor in a given experiment is designated. In this orthogonal array, there are two levels for each factor. While the natures of the factors and the specific values assigned to the levels vary, the selected two levels for each factor are designated by level symbols, shown as the numbers 1 and 2. It is to be understood that the level symbols represent levels according to the particular nature of the factors. For instance, for a factor representing a temperature utilized in the process, the level symbols 1 and 2 represent two different temperature levels. For another factor representing a length of time for a given step of the process, the level symbols 1 and 2 represent two different lengths of time, and so forth. The level symbols are arbitrarily assigned to the factor levels, i.e., there is no requirement that ascending level symbol integers correspond with ascending factor levels, descending factor levels, etc.
Orthogonality, with respect to arrays which define industrial experiments, is defined by Taguchi as follows. For any two columns, representing any two of the factors, it is true that all possible ordered pairs of level symbols are represented in equal numbers over the eight defined experiments. For instance, for factors 1 and 7, the first and fourth experiments use the ordered pair of levels (1, 1), the second and third experiments use the ordered pair of levels (1, 2), the sixth and seventh experiments use the ordered pair of levels (2, 1), and the fifth and eighth experiments use the ordered pair of levels (2, 2). Each of the four possible ordered pairs of levels occur in two of the experiments. The same is true for all of the other possible pairs of factors. Therefore, the array is orthogonal. An orthogonal array is additionally described as symmetric if, as shown in FIG. 1, each of the factors has the same number of levels.
Experimentation based on experiments defined in an orthogonal array provide results which are well suited for identifying optimal levels. Because of the property of orthogonality, a given variance in results between experiments may be analyzed in terms of the variation in levels for the different factors, and those factors which are particularly critical in determining the result can be identified. If, on the other hand, the array were non-orthogonal, it would be more difficult to draw a correlation between the results and the levels for the various factors.
Interactions between factors, as well as manifestations of the factors themselves, can influence which levels for the factors produce optimal results in the industrial process. However, an interaction between two or more factors, each of which is separately given in the orthogonal array, is constrained by the values already assigned to the factors making up the interaction.
In an experimental method in which experiments are defined in terms of levels for factors, there is a drawback, in that a very large number of combinations of levels for all of the factors must be tried before optimal levels can be determined. In the worst case, the total number of experiments equals the product of the numbers of possible levels for all of the factors which go into the process. It is desirable to identify levels for the process factors which produce optimal results while minimizing the total number of experiments which must be performed, in order to save time and cost.
Therefore, it is possible to select a subset of all of the possible permutations for levels for the factors in an industrial process. In FIG. 1, for instance, there are a total of 2.sup.7 or 128 possible permutations of levels for the seven factors having two levels each. However, FIG. 1 is a subset of only eight of the permutations. However, Taguchi does not provide an analytical method for determining when a subset of the total number of permutations of levels defines a set of experiments sufficient to determine the optimum levels for the factors of the industrial process. Therefore, while FIG. 1 is an example of such a subset, it does not show a full set of designed experiments for the industrial process it illustrates.
Another example of the use of orthogonal arrays in the design of experiments in industrial processes is given in Lewandowski et al., "An Automated Method for the Preparation of Orthogonal Arrays for Use in Taguchi Designed Experiments", Computers ind. Engng, Vol. 17, Nos. 1-4, (1989), pp. 502-7. Lewandowski discusses the construction of orthogonal arrays for defining experiments, and covers cases involving mixed level factors (factors having different numbers of levels) and involving interactions. Lewandowski provides a teaching of the general concept of creating an orthogonal array in terms of assigning particular factors to columns. More specifically, on p. 503, lines 36-7 and p, 503, line 50 through p. 504, line 6, it is stated that a factor involved in the greatest number of interactions is placed in column 1, and factors which interact with it are "placed in the appropriate interaction columns . . . based on hueristic rules of interaction."
However, Lewandowski also does not provide a teaching of a method for defining the levels for each of the factors in each of a set of experiments with any certainty that a minimum necessary number of experiments are produced for obtaining experimental results which determine the optimum levels for the factors. Therefore, there remains an unsatisfied need for a method for designing a set of experiments for an industrial process which provides a satisfactory analysis of what levels for the factors give optimal results, but which does not include an unnecessarily large number of experiments.